IRJET- On the Pellian Like Equation 5x2-7y2=-8

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On the Pellian Like Equation 5 x 2  7 y 2  8 S. Vidhayalakshmi1, A. Sathya2, S. Nivetha3 1Professor,

Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India.

Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India.

3PG Scholor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India.

----------------------------------------------------------------------------***------------------------------------------------------------------2Assisant

Abstract – The binary quadratic equation represented by the

X 2  35T 2  4

pellian like equation 5x  7 y  8 is analyzed for its distinct integer solutions. A few interesting relations among the solutions are given. Employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas and parabolas.

2

2

whose smallest positive integer solution is

X 0  12

X 2  35T 2  1

1. INTRODUCTION

~ ~ ( X 0 , To )  (6,1) The general solution of (4) is given by

~ Tn 

This communication concerns with the problem of obtaining non-zero distinct integer solutions to the binary quadratic

 n1  6  35 n1 n 1 n 1 gn  6  35   6  35  Applying Brahmagupta lemma between ( X 0 , T0 ) and ~ ~ ( X n , Tn ) the other integer solutions of (3) are given by

The Diophantine Equation representing the binary quadratic equation to be solved for its non-zero distinct integral solution is

X n 1  6 f n  35 g n Tn 1  f n 

(1)

(2)

35

xn 1  13 f n 

From (1) and (2), we have

Impact Factor value: 7.211

6

gn

    

(5)

From (2), (4) and (5) the values of x and y satisfying (1) are given by

Consider the linear transformations

|

~ 1 gn , X n  fn 2

f n  6  35

2. Method of Analysis

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1 2 35

where

equation given by 5x 2  7 y 2  8 representing hyperbola. A few interesting relations among its solutions are presented.

Knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbolas and parabolas are presented. Also, employing the solutions of the given equation, special Pythagorean triangle is constructed.

y  X  5T

(4)

whose smallest positive integer solution is

The binary quadratic Diophantine equation of the form ax 2  by 2  N , a, b, N  0 are rich in variety and have been analyzed by many mathematicians for their respective integer solutions for particular values of a, b and N . In this context, one may refer [1-14].

x  X  7T

T0  2

To obtain the other solutions of (3), consider the pellian equation is

Key Words: Binary quadratic, Hyperbola, Parabola, Pell equation, Integer solutions.

5x 2  7 y 2  8

(3)

|

77 35

gn

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yn 1  11 f n 

35

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gn

 

The recurrence relation satisfied by the solution x and y are given by

 

xn3 12 xn 2  xn1  0



yn3 12 yn 2  yn1  0



 

Some numerical examples of xn and yn satisfying (1) are given in the Table :1 below

2. Each of the following expressions represents a nasty number:

Table: 1 Numerical Examples

n

xn

yn

0

26

22

1

310

262

2

3694

3122

3

44018

37202

4

524522

443302

5

6250246

5282422

6

74478430

62945762

7

887490914

750066722

From the above table, we observe some interesting relations among the solutions which are presented below: 

xn and yn values are always even.

1. Relation among the solutions are given below:           

xn 2  7 yn1  6 xn1  0 xn3  84 yn1  71xn1  0 yn 2  6 yn1  5xn1  0 yn3  71yn1  60 xn1  0 xn 3  12 xn  2  xn 1  0 xn1  6 xn 2  7 yn 2  0 xn3  xn1  14 yn 2  0 7 yn3  71xn 2  6xn1  0 xn1  71xn3  84 yn3  0 xn3  6 xn 2  7 yn 2  0 71yn  2  5xn 1  6 yn 3  0

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7 yn1  71xn2  6 xn3  0 yn 1  5xn  2  6 yn  2  0 yn1  10 xn 2  yn3  0 yn 2  5xn3  6 yn3  0 71yn 2  6 yn1  5xn3  0 yn1  60xn3  71yn3  0 yn1  12 yn 2  yn3  0 7 yn3  6 xn3  xn 2  0 yn3  6 yn 2  5xn 2  0

|



377 y2n  2  65x2n  2  4



3 77 x2n  3  917 x2n  2  28 7



1 77 x2n  4  10927 x2n  2  336 28



1 77 y2n  3  775x2n  2  24 2



3 77 y2n  4  9235x2n  2  284 71



1 917 y2n  2  65x2n  3  24 2



3 10927 y2n  2  65x2n  4  284 71



3155 y2n  2  13 y2n  3  4



1 1847 y2n  2  13 y2n  4  48 4



3131x2n  4  1561x2n  3  4



3917 y2n  3  775x2n  3  4



1 917 y2n  4  9235x2n  3  24 2



1 10927 y2n  3  775x2n  4  24 2

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

310927 y2n  4  9235x2n  4  4



31847 y2n  3  155 y2n  4  4

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3. Each of the following expressions represents a cubical integer:



1 10927 y3n  4  775 x3n  5  32781yn  2    12   2325 xn  3 



1 10927 y3n  5  9235 x3n  5  32781yn  3    2   27705 xn  3 



1 1847 y3n  4  155 y3n  5  5541yn  2    2   465 yn  3 



1 77 y3n  3  65x3n  3  231yn 1  195xn 1  2



1  77 x3n  4  917 x3n  3  231xn  2    14   2751xn 1 



1  77 x3n  5  10927 x3n  3  231xn  3    168   32781xn 1 



1  77 y4n  4  65 x4n  4  308 y2n  2    2   260 x2n  2  12 



1  77 y3n  4  775 x3n  3  231yn  2    12   2325 xn 1 



1  77 y4n  5  917 x4n  4  308 x2n  3    14   3668 x2n  2  84 



1  77 y3n  5  9235 x3n  3  231yn  3    142   27705 xn 1 



1  77 y4n  6  10927 x4n  4  308 x2n  4    168   43708 x2n  2  1008 



1  917 y3n  3  65 x3n  4  2751yn 1    12   195 xn  2 



1  77 y4n  5  775 x4n  4  308 y2n  3    12   3100 x2n  2  72 



1 10927 y3n  3  65 x3n  5  32781yn 1    142   195 xn  3 



1  77 y4n  6  9235 x4n  4  308 y2n  4    142   36940 x2n  2  852 



1 155 y3n  3  13 y3n  4  465 yn 1  39 yn  2  2



1  917 y4n  4  65 x4n  5  3668 y2n  2    12   260 x2n  3  72 



1 1847 y3n  3  13 y3n  5  5541yn 1  39 yn  3  24



1 10927 y4n  4  43708 y2n  2    142   65 x4n  6  260 x2n  4  852 



1 131x3n  5  1561x3n  4  393xn  3  4683xn  2  2



1 155 y4n  4  13x4n  5  620 y2n  2    2   52 y2n  3  12 



1  917 y3n  4  775 x3n  4  2751yn  2    2   2325 xn  2 



1 1847 y4n  4  13 y4n  6  7388 y2n  2    24   52 y2n  4  144 



1  917 y3n  5  9235 x3n  4  2751yn  3    12   27705 xn  2 



1  917 x4n  6  10927 x4n  5  3668 x2n  4    14   43708 x2n  3  84 

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Impact Factor value: 7.211

4. Each of the following expressions represents a biquadratic integer:

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

1  917 y4n  5  775 x4n  5  3668 y2n  3    2   3100 x2n  3  12 



1 1847 y5n  5  13 y5n  7  9235 y3n  3    24   65 y3n  5  18470 yn 1  130 yn  3 



1  917 y4n  6  9235 x4n  5  3668 y2n  4    12   36940 x2n  3  72 



 1  917 x5n  7  10927 x5n  6  4585 x3n  5   14   54635 x3n  4  9170 xn  3  109270 xn  2 



1 10927 y4n  5  775 x4n  6  43708 y2n  3    12   3100 x2n  4  72 



1  917 y5n  6  775 x5n  6  4585 y3n  4    2   3875 x3n  4  9170 yn  2  7750 xn  2 



1 10927 y4n  6  9235 x4n  6  43708 y2n  4    2   36940 x2n  4  12 



 1  917 y5n  7  9235 x5n  6  4585 y3n  5   12   46175 x3n  4  9170 yn  3  923500 xn  2 



1 1847 y4n  5  155 y4n  6  7388 y2n  3    2   620 y2n  4  12 



1 10927 y5n  6  775 x5n  7  54635 y3n  4    12   3875 x3n  5  109270 yn  2  7750 xn  3 



1 10927 y5n  7  9235 x5n  7  54635 y3n  5    2   46175 x3n  5  109270 yn  3  92350 xn  3 



1 1847 y5n  6  155 y5n  7  9235 y3n  4    2   775 y3n  5  18470 yn  2  1550 yn  3 

5. Each of the following expressions represents a quintic integer: 

1  77 y5n  5  65 x5n  5  385 y3n  3  325 x3n  3    2   770 yn 1  650 xn 1 



 1  77 x5n  6  917 x5n  5  385 x3n  4   14   4585 x3n  3  770 xn  2  9170 xn 1 



 1  77 x5n  7  10927 x5n  5  385 y3n  5   168   54635 x3n  3  770 xn  3  109270 xn 1 



1  77 y5n  6  775 x5n  5  385 y3n  4    12   3875 x3n  3  770 yn  2  7750 xn 1 



 1  77 y5n  7  9235 x5n  5  385 y3n  5   142   46175 x3n  3  770 yn  3  92350 xn 1 







REMARKABLE OBSERVATIONS I. Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of hyperbolas which are presented in Table: 2 below: Table: 2 Hyperbolas S.NO

Hyperbola

1

Y 2  35 X 2  16

2

Y 2  35 X 2  784

1  917 y5n  5  65 x5n  6  4585 y3n  3    12   325 x3n  4  9170 yn 1  650 xn  2 

3

Y 2  35 X 2  112896

1 10927 y5n  5  65 x5n  7  54635 y3n  3    142   325 x3n  5  109270 yn 1  650 xn  3 

4

Y 2  35 X 2  576

5

Y 2  35 X 2  80656

1 155 y5n  5  13 y5n  6  775 y3n  3    2   65 y3n  4  1550 yn 1  130 yn  2 

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Impact Factor value: 7.211

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(X,Y)

11xn 1  13 yn 1 ,   77 yn 1  65 xn 1

  

155 xn 1  13xn  2 ,     77 xn  2  917 xn 1  1847 xn 1  13xn 3 ,   77 xn 3  10927 xn 1

  

131xn 1  13 yn 1 ,     77 yn  2  775 xn 1  1561xn 1  13 yn  3 ,   77 yn 3  9235 xn 1

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6

Y 2  35 X 2  576

11xn  2  155 yn 1 ,     917 yn 1  65 xn  2 

4

12Y  35 X 2  576

7

Y 2  35 X 2  80656

11xn 3  1847 yn 1 ,     10927 yn 1  65 xn 3 

5

 142Y  35 X 2  80656 1561xn 1  13 yn 3 ,   77 y  9235 x  284 2n  4 2n  2  

8

Y 2  35 X 2  400

11yn  2  131yn 1 ,   775 yn 1  65 yn  2

  

6

12Y  35 X 2  576

9

Y 2  35 X 2  57600

11yn 3  1561yn 1 ,     9235 yn 1  65 yn 3 

7

 142Y  35 X 2  80656 11xn 3  1847 yn 1 ,   10927 y 2n  2  65 x2n  4  284  

10

Y 2  35 X 2  784

1847 xn  2  155 xn 3 ,     917 xn  3  10927 xn  2 

8

2Y  7 X 2  80

131xn  2  155 yn  2 ,     917 yn  2  775 xn  2 

11yn  2  131yn 1 ,     775 y2n  2  65 y2n 3  20 

9

120Y  7 X 2  11520

11yn 3  1561yn 1 ,    1847 y2n  2  13 y2n  4  48 

11

12

Y 2  35 X 2  16

Y 2  35 X 2  576

1561xn  2  155 yn 3 ,   917 yn 3  9235 xn  2

  

13

Y 2  35 X 2  576

131xn 3  1847 yn  2 ,     10927 yn  2  775 xn 3 

14

Y 2  35 X 2  16

1561xn 3  1847 yn 3 ,     10927 yn 3  9235 xn 3 

15

Y 2  35 X 2  400

131yn 3  1561yn  2 ,     9235 yn  2  775 yn 3 

131xn 1  13 yn 1 ,    77 y  775 x  24 2n  3 2n  2  

11xn  2  155 yn 1 ,     917 y2n  2  65 x2n  3  24 

10

2Y  5 X 2  112

1847 xn  2  155 xn  3 ,     917 x2n  4  10927 x2n  3  28 

11

2Y  35 X 2  16

131xn  2  155 yn  2 ,    917 y  775 x  4 2n  3 2n  3  

12

12Y  35 X 2  576

1561xn  2  155 yn 3 ,    917 y  9235 x  24 2n  4 2n  3  

II. Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of parabolas which are presented in Table: 3 below:

13

12Y  35 X 2  576

131xn 3  1847 yn  2 ,     10927 y2n 3  775 x2n  4  24 

Table: 3 Parabolas

14

2Y  35 X 2  16

1561xn 3  1847 yn 3 ,    10927 y2n  4  9235 x2n  4  4 

15

2Y  7 X 2  80

131yn  3  1561yn  2 ,     9235 y2n 3  775 y2n  4  20 

S.NO

Parabola

(X,Y)

1

2Y  35 X 2  16

11xn 1  13 yn 1 ,    77 y  65 x  4 2n  2 2n  2  

2

2Y  5 X 2  112

155 xn 1  13xn  2 ,    77 x  917 x  28 2n  3 2n  2  

3

24Y  5 X  16128

1847 xn 1  13xn  3 ,     77 x2n  4  10927 x2n  2  336 

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III. Let p, q be two non-zero distinct integers such that p  q  0 . Treat p, q as the generators of the Pythagorean triangle T ( ,  ,  )

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where   2 pq ,   p 2  q 2 ,   p 2  q 2 , p  q  0 Taking p  xn 1  yn 1 , q  yn 1 , it is observed T ( ,  ,  ) is satisfied by the following relations: 

3  7  10  16



4

A     p



2

A  xn 1 yn 1 p

[7]

S. Vidhyalakshmi,et al., Observations on the hyperbola ax 2  a  1y 2  3a  1 , Discovery, 4(10) (2013) 22-24.

[8]

M.A. gopalan,et al., Integral points on the hyperbola 2  ay 2  4ak  1  2k 2 , a, k  0 , Indian Journal of science, 1(2) (2012) 125-126.

[9]

M.A. Gopalan and S. Vidhyalakshmi and A. Kavitha, on the integer solutions of binary quadratic equation, x 2  4k 2  1y 2  4t , k , t  0 , BOMSR, 2(2014) 42-46.

[10]

T.R. Usha Rani and K. Ambika, Observation on the NonHomogeneous Binary Quadratic Diophantine Equation 5x 2  6 y 2  5 , Journal of Mathematics and Informatics, Vol-10,Dec (2017), 67-74.

[11]

M.A. Gopalan and Sharadha Kumar, On the Hyperbola 2 x 2  3 y 2  23 , Journal of Mathematics and Informatics, vol-10,Dec (2017),1-9.

[12]

M.A.Gopalan, S.Vidhyalakshmi and A. Kavitha, Observationson the Hyperbola 10y 2  3x 2  13, Archimedes J. Math., 3(1) (2013), 31-34.

[13]

M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, On the integral solutions of the binary quadratic equation x 2  15y 2  11t , Scholars Journal of Engineering and Technology, 2(2A) (2014), 156-158.

[14]

Shreemathi Adiga, N. Anusheela and M.A. Gopalan, Observations on the Positive Pell Equation y 2  20x 2  1 , International Journal of Pure and Applied Mathematics, 120(6) (2018), 11813-11825

that

Where A, P represent the area and perimeter of T ( ,  ,  ) .

3. CONCLUSION In this paper, we have presented infinitely many integer solutions for the Pellian like equation 5x 2  7 y 2  8 . As the binary quadratic Diophantine equations are rich in variety, one may search for the other choices of Pell equations and determine their integer solutions along with suitable properties.

REFERENCES [1]

R.D. Carmichael,The Theory of Numbers and Diophantine Analysis, Dover Publications, New York (1950).

[2]

L.E. Disckson, History of Theory of Numbers, vol II, Chelsea publishing Co., New York (1952).

[3]

L.J. Mordell, Diophantine Equations, Academic press, London (1969).

[4]

M.A. Gopalan and R. Anbuselvi, Integral solutions of 4ay 2  a  1x 2  3a  1 , Acta Ciencia Indica, XXXIV(1) (2008) 291-295.

[5]

M.A. Gopalan,et al., Integral points on the hyperbola a  2x2  ay 2  4ak 1  2k 2 , a, k  0 , Indian Journal of Science, 1(2) (2012) 125-126.

[6]

M.A. Gopalan, S. Devibala and R. Vidhylakshmi, Integral points on the hyperbola 2 x 2  3 y 2  5 , Amearican Jornal of Applied Mathematics and Mathematical Sciences, 1(2012) 1-4.

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a  2x

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